A general model of incompatible linearized elasticity is presented and analyzed, based on the lin... more A general model of incompatible linearized elasticity is presented and analyzed, based on the linearized strain and its associated incompatibility tensor field. Elastic strain incompatibility accounts for the presence of dislocations, whose motion is ultimately responsible for the plastic behaviour of solids. The specific functional setting is built up, on which existence results are proved. Our solution strategy is essentially based on the projection of the governing equations on appropriate subspaces in the spirit of Leray decomposition of solenoidal square-integrable velocity fields in hydrodynamics. It is also strongly related with the Beltrami decomposition of symmetric tensor fields in the wake of previous works by the authors. Moreover a novel model parameter is introduced, the incompatibility modulus, that measures the resistance of the elastic material to incompatible deformations. An important result of our study is that classical linearized elasticity is recovered as the ...
This paper is devoted to a numerical implementation of the FrancfortMarigo model of damage evolut... more This paper is devoted to a numerical implementation of the FrancfortMarigo model of damage evolution in brittle materials. This quasi-static model is based, at each time step, on the minimization of a total energy which is the sum of an elastic energy and a Griffith energy release rate. Such a minimization is carried over all geometric mixtures of the two, healthy and damaged, elastic phases, respecting an irreversibility constraint. Numerically, we consider a situation where two well separated phases coexist, and model their interface by a level set function that is transported according to the shape derivative of the minimized total energy. In the context of interface variations (Hadamard method) and using a steepest descent algorithm, we compute local minimizers of this quasistatic damage model. Initially, the damaged zone is nucleated by using the so-called topological derivative. We show that, when the damaged phase is very weak, our numerical method is able to predict crack pr...
This paper deals with a novel hydraulic fracture model based on the concept of topological deriva... more This paper deals with a novel hydraulic fracture model based on the concept of topological derivative. The basic idea consists in adapting the Francfort-Marigo damage model to the context of hydraulic fracture. The Francfort-Marigo damage model is a variational approach to describe the behavior of brittle materials under the quasi-static loading assumption, focusing on the evolution of damage regions under an irreversibility constraint. In our problem, the loading comes out from a prescribed pressure acting within the damage region, which is used to trigger the hydraulic fracturing process. A shape functional given by the sum of the total potential energy of the system with a Griffith-type dissipation energy term is minimized with respect to a set of ball-shaped pressurized inclusions by using the topological derivative concept. In particular, the topological asymptotic expansion of the shape functional with respect to the nucleation of a circular inclusion endowed with non-homogene...
The process of fluid-driven crack propagation in permeable rocks is investigated by a simple hydr... more The process of fluid-driven crack propagation in permeable rocks is investigated by a simple hydro-mechanical model and the concept of topological derivative. Analytical and numerical computations are made to propose a promising model tested and validated on a series of bidimensional benchmark examples. The main guidelines for this model is the use of simple finite elements with a minimal number of user-defined algorithmic parameters.
The topological derivative is defined as the first term of the asymptotic expansion of a given sh... more The topological derivative is defined as the first term of the asymptotic expansion of a given shape functional with respect to a small parameter that measures the size of a singular domain perturbation. It has applications in many different fields such as shape and topology optimization, inverse problems, image processing and mechanical modeling including synthesis and/or optimal design of microstructures, fracture mechanics sensitivity analysis and damage evolution modeling. The topological derivative has been fully developed for a wide range of second order differential operators. In this paper we deal with the topological asymptotic expansion of a class of shape functionals associated with elliptic differential operators of order 2m, m ≥ 1. The general structure of the polarization tensor is derived and the concept of degenerate polarization tensor is introduced. We provide full mathematical justifications for the derived formulas, including precise estimates of remainders.
We prove the existence of minimizers for functionals defined over the class of convex domains con... more We prove the existence of minimizers for functionals defined over the class of convex domains contained inside a bounded set D of R^N and with prescribed volume. Some applications are given, in particular we prove that the eigenvalues of differential operators of second and fourth order with non-constant coefficients as well as integral functionals depending on the solution of an elliptic equation can be minimized over this class of domains. Another application of this result is related to the famous Newton problem of minimal resistance. In general, all the results we shall develop hold for elliptic operators of any even order larger that 0.
A novel model of elasto-plasticity based on an intrinsic approach is proposed. The model variable... more A novel model of elasto-plasticity based on an intrinsic approach is proposed. The model variables are the linearized strain and its incompatibility. Elastic strain incompatibility accounts for the presence of dislocations in the microstructure, which are responsible for the plastic behaviour of solids. The functional analysis setting is built up, on which existence results are proved.
Considering the existence of solutions to a minimum problem for dislocations in finite elasticity... more Considering the existence of solutions to a minimum problem for dislocations in finite elasticity [21], in the present paper we first exhibit a constraint reaction field, due to the geometrical constraint that the deformation curl is a concentrated measure. The appropriate functional spaces and their properties needed to describe dislocations are then established. Following the preceding theoretical developments, the first variation of the energy at the minimum points with respect to Lipschitz variations of the lines and to curl-free deformations is carried on. Our first purpose is to show that the constraint reaction provides explicit expressions of the Piola-Kirchhoff stress and Peach-Köhler force. Then, equilibrium at optimality shows that the latter force is balanced by a defect-induced configurational force. Our main result is to establish that the Peach-Köhler force is a concentrated Radon measure in the dislocation. In the modeling application, in order to consider complex st...
This paper deals with a variational problem for dislocations in which the curl of the deformation... more This paper deals with a variational problem for dislocations in which the curl of the deformation tensor is constrained by a concentrated measure in a set of lines, called the dislocation density, while the energy density involves the deformation tensor and its gradient, specifically, the curl and the divergence in two distinct terms. To solve this constrained variational problem in finite elasticity, the notion of integral 1-current is used in the spirit of previous work by the same authors. No assumptions on the lines are made except the classical requirement to be closed loops or end at the crystal boundary. Since the displacement field is by essence multiple valued, it is chosen to work with torus-valued maps. Moreover, graphs of harmonics maps are at the heart of such a problem, and therefore our theory is grounded in an analysis of their properties with a view to dislocation modeling. Our main result shows that dislocation density and displacement graph boundary are bound noti...
PurposeIn the paper an approach for crack nucleation and propagation phenomena in brittle plate s... more PurposeIn the paper an approach for crack nucleation and propagation phenomena in brittle plate structures is presented.Design/methodology/approachThe Francfort–Marigo damage theory is adapted to the Kirchhoff and Reissner–Mindlin plate bending models. Then, the topological derivative method is used to minimize the associated Francfort–Marigo shape functional. In particular, the whole damaging process is governed by a threshold approach based on the topological derivative field, leading to a notable simple algorithm.FindingsNumerical simulations are driven in order to verify the applicability of the proposed method in the context of brittle fracture modeling on plates. The obtained results reveal the capability of the method to determine nucleation and propagation including bifurcation of multiple cracks with a minimal number of user-defined algorithmic parameters.Originality/valueThis is the first work concerning brittle fracture modeling of plate structures based on the topologica...
A general model of incompatible linearized elasticity is presented and analyzed, based on the lin... more A general model of incompatible linearized elasticity is presented and analyzed, based on the linearized strain and its associated incompatibility tensor field. Elastic strain incompatibility accounts for the presence of dislocations, whose motion is ultimately responsible for the plastic behaviour of solids. The specific functional setting is built up, on which existence results are proved. Our solution strategy is essentially based on the projection of the governing equations on appropriate subspaces in the spirit of Leray decomposition of solenoidal square-integrable velocity fields in hydrodynamics. It is also strongly related with the Beltrami decomposition of symmetric tensor fields in the wake of previous works by the authors. Moreover a novel model parameter is introduced, the incompatibility modulus, that measures the resistance of the elastic material to incompatible deformations. An important result of our study is that classical linearized elasticity is recovered as the ...
This paper is devoted to a numerical implementation of the FrancfortMarigo model of damage evolut... more This paper is devoted to a numerical implementation of the FrancfortMarigo model of damage evolution in brittle materials. This quasi-static model is based, at each time step, on the minimization of a total energy which is the sum of an elastic energy and a Griffith energy release rate. Such a minimization is carried over all geometric mixtures of the two, healthy and damaged, elastic phases, respecting an irreversibility constraint. Numerically, we consider a situation where two well separated phases coexist, and model their interface by a level set function that is transported according to the shape derivative of the minimized total energy. In the context of interface variations (Hadamard method) and using a steepest descent algorithm, we compute local minimizers of this quasistatic damage model. Initially, the damaged zone is nucleated by using the so-called topological derivative. We show that, when the damaged phase is very weak, our numerical method is able to predict crack pr...
This paper deals with a novel hydraulic fracture model based on the concept of topological deriva... more This paper deals with a novel hydraulic fracture model based on the concept of topological derivative. The basic idea consists in adapting the Francfort-Marigo damage model to the context of hydraulic fracture. The Francfort-Marigo damage model is a variational approach to describe the behavior of brittle materials under the quasi-static loading assumption, focusing on the evolution of damage regions under an irreversibility constraint. In our problem, the loading comes out from a prescribed pressure acting within the damage region, which is used to trigger the hydraulic fracturing process. A shape functional given by the sum of the total potential energy of the system with a Griffith-type dissipation energy term is minimized with respect to a set of ball-shaped pressurized inclusions by using the topological derivative concept. In particular, the topological asymptotic expansion of the shape functional with respect to the nucleation of a circular inclusion endowed with non-homogene...
The process of fluid-driven crack propagation in permeable rocks is investigated by a simple hydr... more The process of fluid-driven crack propagation in permeable rocks is investigated by a simple hydro-mechanical model and the concept of topological derivative. Analytical and numerical computations are made to propose a promising model tested and validated on a series of bidimensional benchmark examples. The main guidelines for this model is the use of simple finite elements with a minimal number of user-defined algorithmic parameters.
The topological derivative is defined as the first term of the asymptotic expansion of a given sh... more The topological derivative is defined as the first term of the asymptotic expansion of a given shape functional with respect to a small parameter that measures the size of a singular domain perturbation. It has applications in many different fields such as shape and topology optimization, inverse problems, image processing and mechanical modeling including synthesis and/or optimal design of microstructures, fracture mechanics sensitivity analysis and damage evolution modeling. The topological derivative has been fully developed for a wide range of second order differential operators. In this paper we deal with the topological asymptotic expansion of a class of shape functionals associated with elliptic differential operators of order 2m, m ≥ 1. The general structure of the polarization tensor is derived and the concept of degenerate polarization tensor is introduced. We provide full mathematical justifications for the derived formulas, including precise estimates of remainders.
We prove the existence of minimizers for functionals defined over the class of convex domains con... more We prove the existence of minimizers for functionals defined over the class of convex domains contained inside a bounded set D of R^N and with prescribed volume. Some applications are given, in particular we prove that the eigenvalues of differential operators of second and fourth order with non-constant coefficients as well as integral functionals depending on the solution of an elliptic equation can be minimized over this class of domains. Another application of this result is related to the famous Newton problem of minimal resistance. In general, all the results we shall develop hold for elliptic operators of any even order larger that 0.
A novel model of elasto-plasticity based on an intrinsic approach is proposed. The model variable... more A novel model of elasto-plasticity based on an intrinsic approach is proposed. The model variables are the linearized strain and its incompatibility. Elastic strain incompatibility accounts for the presence of dislocations in the microstructure, which are responsible for the plastic behaviour of solids. The functional analysis setting is built up, on which existence results are proved.
Considering the existence of solutions to a minimum problem for dislocations in finite elasticity... more Considering the existence of solutions to a minimum problem for dislocations in finite elasticity [21], in the present paper we first exhibit a constraint reaction field, due to the geometrical constraint that the deformation curl is a concentrated measure. The appropriate functional spaces and their properties needed to describe dislocations are then established. Following the preceding theoretical developments, the first variation of the energy at the minimum points with respect to Lipschitz variations of the lines and to curl-free deformations is carried on. Our first purpose is to show that the constraint reaction provides explicit expressions of the Piola-Kirchhoff stress and Peach-Köhler force. Then, equilibrium at optimality shows that the latter force is balanced by a defect-induced configurational force. Our main result is to establish that the Peach-Köhler force is a concentrated Radon measure in the dislocation. In the modeling application, in order to consider complex st...
This paper deals with a variational problem for dislocations in which the curl of the deformation... more This paper deals with a variational problem for dislocations in which the curl of the deformation tensor is constrained by a concentrated measure in a set of lines, called the dislocation density, while the energy density involves the deformation tensor and its gradient, specifically, the curl and the divergence in two distinct terms. To solve this constrained variational problem in finite elasticity, the notion of integral 1-current is used in the spirit of previous work by the same authors. No assumptions on the lines are made except the classical requirement to be closed loops or end at the crystal boundary. Since the displacement field is by essence multiple valued, it is chosen to work with torus-valued maps. Moreover, graphs of harmonics maps are at the heart of such a problem, and therefore our theory is grounded in an analysis of their properties with a view to dislocation modeling. Our main result shows that dislocation density and displacement graph boundary are bound noti...
PurposeIn the paper an approach for crack nucleation and propagation phenomena in brittle plate s... more PurposeIn the paper an approach for crack nucleation and propagation phenomena in brittle plate structures is presented.Design/methodology/approachThe Francfort–Marigo damage theory is adapted to the Kirchhoff and Reissner–Mindlin plate bending models. Then, the topological derivative method is used to minimize the associated Francfort–Marigo shape functional. In particular, the whole damaging process is governed by a threshold approach based on the topological derivative field, leading to a notable simple algorithm.FindingsNumerical simulations are driven in order to verify the applicability of the proposed method in the context of brittle fracture modeling on plates. The obtained results reveal the capability of the method to determine nucleation and propagation including bifurcation of multiple cracks with a minimal number of user-defined algorithmic parameters.Originality/valueThis is the first work concerning brittle fracture modeling of plate structures based on the topologica...
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